In conventional turbomachines, a lining element, for example an inner liner, is fastened on a support element, and also a membrane seal slot for a membrane seal is formed on the support element. The support element is surrounded, for example, with a compressor final temperature during operation of the turbomachine and therefore makes no really high demands on the oxidation resistance of the material. In most cases, therefore, it is sufficient if the support elements are manufactured for example from Stg50T, Stg41T, St530TS or from GGG40. In order to also be able to reduce a cooling air portion of a turbine platform in the case of such turbomachines, this platform is constructed as short as possible, as a result of which, however, the static pressure in front of the leading edge of the turbine blade increases owing to the aerodynamic resistance of a turbine blade, and causes a so-called bow wave effect. As a result of the higher static pressure in the region of the blade leading edge, the penetration of hot gas in the gap between a combustion chamber and the turbine platform can occur in turn. The parts which are exposed to impingement by hot gas, such as the inner liner or the membrane seal, are customarily manufactured from a nickel-based alloy and are therefore adequately protected even without a thermal barrier coating. On the other hand, the support elements, however, oxidize relatively intensely on account of the high temperatures which are created as a result of the hot gas penetration. In order to avoid this, a barrier coating, for example an Inconel 625 coating, is customarily deposited onto the support elements, which, however, is time-consuming to produce and gives rise to high costs. In addition, the gap across the membrane seal on the platform side is purged with cooling air. Despite these measures, however, abrasive wear on the support element in the region of an axial end face in the gap occurs, which in the worst case can lead to increased leakage up to the point of a disintegration of the sealing membrane.